The Hubbard-Stratonovitch auxiliary-field approach to projection of the ground state of an interacting fermion system from a trial state is examined. It is shown that the method is equivalent to solving a differential equation with diffusion, drift, and branching terms on the manifold of normalized Slater determinants. Explicit general expressions are given for the coefficients of the diffusion equation in terms of the Hubbard-Stratonovitch fields. The form of the equation is somewhat similar to that obtained in the continuum Green’s-function Monte Carlo method, although its interpretation and relation to the physical many-body problem is quite different. The character of the representation of the many-body ground state arising in the auxiliary-field projection approach is discussed within this framework. The diffusion process is normally found to concentrate the states representing the ground state near the classical mean-field solutions. The consequences of this picture of the auxiliary-field approach for the ‘‘fermion minus-sign’’ problem in this context are discussed, and some conclusions are reached concerning the range of validity of a recently suggested approximation where minus signs are ignored.